Mathematicians Reveal Secrets Of The Ancient And Universal Art Of Symmetry

Source:

http://www.sciencedaily.com/releases/2008/05/080521173851.htm

ScienceDaily (May 24, 2008) — Humans have used symmetrical patterns for thousands of years in both functional and decorative ways. Now, a new book by three mathematicians offers both math experts and enthusiasts a new way to understand symmetry and a fresh way to see the world.

In The Symmetries of Things, eminent Princeton mathematician John H. Conway teams up with Chaim Goodman-Strauss of the University of Arkansas and Heidi Burgiel of Bridgewater State College to present a comprehensive mathematical theory of symmetry in a richly illustrated volume. The book is designed to speak to those with an interest in math, artists, working mathematicians and researchers.
“Symmetry and pattern are fundamentally human preoccupations in the same way that language and rhythm are. Any culture that is making anything has ornament and is preoccupied with this visual rhythm,” Goodman-Strauss said. “There are actually Neolithic examples of many of these patterns. The fish-scale pattern, for example, is 22,000 years old and shows up all over the world in all kinds of contexts.”
Symmetrical objects and patterns are everywhere. In nature, there are flowers composed of repeating shapes that rotate around a central point. Architects trim buildings with friezes that repeat design elements over and over.
Mathematicians, according to Goodman-Strauss, are latecomers to the human fascination with pattern. While mathematicians bring their own particular concerns, “we’re also able to say things that other people might not be able to say.”
Symmetries of Things contributes a new system of notation or descriptive categories for symmetrical patterns and a host of new proofs. The first section of the book is written to be accessible to a general reader with an interest in the subject. Sections two and three are aimed at mathematicians and experts in the field. The entire book, Goodman-Strauss said, “is meant to be engaging and reveal itself visually as well.”
To explain the significance within mathematics of understanding symmetry, Goodman-Strauss began by talking about mathematics in general: “Mathematicians above anything else study structure, structure for its own sake, mental structure, not necessarily physical structure. That’s why mathematics is so good at describing the world. What more fundamental kind of structure could you consider than the way patterns can be laid out in a plane?”
While mathematics may be called “a descriptive art,” Goodman-Strauss noted that mathematicians are not simply trying to describe. Rather, he said, “We’re trying to understand what inherently can be described in a quantitative, analytical way.”
For about a hundred years, mathematicians have used a system developed by crystallographers to describe symmetries, a system that didn’t easily generalize to other situations. Conway developed a notational system that is more useful for mathematicians, a flexible, intuitive system that is “much more than a naming system,” according to Goodman-Strauss.
“Conway is one of the best notation-makers in the world,” Goodman-Strauss said. “A good notation is amazing because it’s not just a way of naming things. It’s a way of making the structure of things transparent and simultaneously providing a way of enumerating them, classifying them and proving that’s what the classification is – all at once. That’s really the big exciting thing.”
The second section of the book discusses the orbifold, which is a tool for understanding symmetries. As the researchers write in the book’s introduction, Goodman-Strauss “had been preaching the gospel of the orbifold signature on his own, and was known for his gorgeous illustrations.”
Orbifolds are formed when symmetrical patterns on a surface are folded or rolled so that every distinct feature, every point on a pattern, is brought together with its corresponding point. The result is a geometrical shape, such as a sphere, a cone or a cylinder, that shows one example of the design element that was repeated to make the symmetrical pattern.
As a tool, the orbifold pattern provides an efficient way to understand patterns. Goodman-Strauss uses the example of the ancient and universal fish-scale pattern.
“Why would the fish scale pattern be so compelling and so interesting and be the basis for all kinds of other patterns? It’s very easy – because it has a very simple orbital,” Goodman-Strauss explained. “You want to get to a simple pattern on an orbifold. When you do that, then the pattern is strong and dynamic.”
Goodman-Strauss also hosts his own Web site at http://www.mathbun.com, featuring examples of symmetrical patterns and orbifolds, along with photos of other math- and art-related projects.

Fausto Intilla - www.oloscience.com

Mathematics Simplifies Sleep Monitoring

Source:

http://www.sciencedaily.com/releases/2008/05/080507105644.htm

ScienceDaily (May 7, 2008) — A UQ researcher has created a new way to measure breathing patterns in sleeping infants which may also work for adults.
The researcher, PhD student Philip Terrill, has created a mathematical formula that measures varying breathing patterns which indicate different sleep states such as active or quiet sleep.
Mr Terrill said a band, placed around the child's chest, recorded breathing rates which were then analysed using the new formula based on the maths of chaos theory.
It has been successfully tested on 30 children so far.
Current sleep monitoring involves an overnight stay in a hospital sleep lab with specialised equipment needing regular attention of a nurse, doctor or sleep technician.
Mr Terrill said he hoped his formula would form the basis of an automated sleep monitoring system that was cheaper and easier to use than current methods. "In the future, diagnosing a sleep problem may be as simple as putting on a breathing monitor during a night's sleep at home, in your own bed," Mr Terrill said.
"This would mean that those children with sleep problems could be quickly diagnosed and treated appropriately."
Minor infant sleeping problems can result in daytime sleepiness and inattention with prolonged problems causing behavioural and learning difficulties.
Mr Terrill said clinical research showed that up to 20 percent of Australian children have symptoms of sleep problems and there were very few facilities available to investigate sleep problems in Queensland children.
He said previous work analysed sleep breathing patterns using conventional statistical methods but his work used techniques from a branch of mathematics called chaos theory. The next step is to test his formula on teenagers and adults.
Adapted from materials provided by University Of Queensland.

Fausto Intilla - www.oloscience.com

Music Has Its Own Geometry, Researchers Find

Source:

http://www.sciencedaily.com/releases/2008/04/080417142454.htm

ScienceDaily (Apr. 18, 2008) — The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.

And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors -- Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."
Like what?
"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."
"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
"Our methods are not so great at distinguishing Aerosmith from the Rolling Stones," Tymoczko said. "But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music."
Adapted from materials provided by Princeton University.

Fausto Intilla - www.oloscience.com